Portfolio Optimization with Alternative Risk Measures

Portfolio Optimization with Alternative Risk Measures Prof. Daniel P. Palomar The Hong Kong University of Science and Technology (HKUST) MAFS6010R- Portfolio Optimization with R MSc in Financial Mathematics Fall 2018-19, HKUST, Hong Kong

Outline 1 Introduction 2 Warm-Up: Markowitz Portfolio Signal model Markowitz formulation Drawbacks of Markowitz portfolio 3 Alternative Measures of Risk: DR, VaR, CVaR, and DD 4 Mean-DR portfolio 5 Mean-CVaR portfolio 6 Mean-DD portfolio 7 Conclusions

Motivation The Markowitz portfolio has never been embraced by practitioners, among other reasons because 1 variance is not a good measure of risk in practice since it penalizes both the unwanted high losses and the desired low losses: the solution is to use alternative measures for risk, e.g., VaR and CVaR, 2 it is highly sensitive to parameter estimation errors (i.e., to the covariance matrix Σ and especially to the mean vector µ): solution is robust optimization, 3 it only considers the risk of the portfolio as a whole and ignores the risk diversification: solution is the risk-parity portfolio. We will here consider more meaningful measures for risk than the variance, like the downside risk (DR), Value-at-Risk (VaR), Conditional VaR (CVaR) or Expected Shortfall (ES), and drawdown (DD). D. Palomar (HKUST) CVaR Portfolio 4 / 58

Returns Let us denote the log-returns of N assets at time t with the vector r t R N. The time index t can denote any arbitrary period such as days, weeks, months, 5-min intervals, etc. F t 1 denotes the previous historical data. Econometrics aims at modeling r t conditional on F t 1. r t is a multivariate stochastic process with conditional mean and covariance matrix denoted as 1 µ t E [r t F t 1 ] Σ t Cov [r t F t 1 ] = E [(r t µ t )(r t µ t ) T F t 1 ]. 1 Y. Feng and D. P. Palomar, A Signal Processing Perspective on Financial Engineering. Foundations and Trends in Signal Processing, Now Publishers, 2016. D. Palomar (HKUST) CVaR Portfolio 7 / 58

I.I.D. Model For simplicity we will assume that r t follows an i.i.d. distribution (which is not very innacurate in general). That is, both the conditional mean and conditional covariance are constant µ t = µ, Σ t = Σ. Very simple model, however, it is one of the most fundamental assumptions for many important works, e.g., the Nobel prize-winning Markowitz portfolio theory 2. 2 H. Markowitz, Portfolio selection, J. Financ., vol. 7, no. 1, pp. 77 91, 1952. D. Palomar (HKUST) CVaR Portfolio 8 / 58

Parameter Estimation Consider the i.i.d. model: r t = µ + w t, where µ R N is the mean and w t R N is an i.i.d. process with zero mean and constant covariance matrix Σ. The mean vector µ and covariance matrix Σ have to be estimated in practice based on T observations. The simplest estimator is the sample estimator: sample mean estimator: ˆµ = 1 T T t=1 r t sample covariance matrix: ˆΣ = 1 T 1 T t=1 (r t ˆµ)(r t ˆµ) T. Many more sophisticated estimators exist, namely: shrinkage estimators, Black-Litterman estimators, etc. D. Palomar (HKUST) CVaR Portfolio 9 / 58

Portfolio Return Suppose the budget is B dollars. The portfolio w R N denotes the normalized weights of the assets such that 1 T w = 1 (then Bw denotes dollars invested in the assets). For each asset, the initial wealth is Bw i and the end wealth is Then the portfolio return is Bw i (p i,t /p i,t 1 ) = Bw i (R it + 1). R p t = Ni=1 Bw i (R it + 1) B B N N = w i R it w i r it = w T r t i=1 i=1 The portfolio expected return and variance are w T µ and w T Σw, respectively. 3 3 G. Cornuejols and R. Tütüncü, Optimization Methods in Finance. Cambridge University Press, 2006. D. Palomar (HKUST) CVaR Portfolio 11 / 58

Performance Measures Expected return: w T µ Volatility: w T Σw Sharpe Ratio (SR): expected return per unit of risk SR = wt µ r f w T Σw where r f is the risk-free rate (e.g., interest rate on a three-month U.S. Treasury bill). Information Ratio (IR): IR = wt µ w T Σw Drawdown (DD): decline from a historical peak of the cumulative profit X(t): D(t) = max 1 τ t X(τ) X(t) (unnormalized) VaR (Value at Risk) ES (Expected Shortfall) or CVaR (Conditional Value at Risk) D. Palomar (HKUST) CVaR Portfolio 12 / 58

Practical Constraints Capital budget constraint: w T 1 = 1. Long-only constraint: Market-neutral constraint: w 0. w T 1 = 0. Turnover constraint: w w 0 1 u where w 0 is the currently held portfolio. D. Palomar (HKUST) CVaR Portfolio 13 / 58

Practical Constraints Holding constraint: l w u where l R N and u R N are lower and upper bounds of the asset positions, respectively. Cardinality constraint: Leverage constraint: w 0 K. w 1 2. D. Palomar (HKUST) CVaR Portfolio 14 / 58

Risk Control In finance, the expected return w T µ is very relevant as it quantifies the average benefit. However, in practice, the average performance is not enough to characterize an investment and one needs to control the probability of going bankrupt. Risk measures control how risky an investment strategy is. The most basic measure of risk is given by the variance 4 : a higher variance means that there are large peaks in the distribution which may cause a big loss. There are more sophisticated risk measures such as downside risk, VaR, ES, drawdown, etc. 4 H. Markowitz, Portfolio selection, J. Financ., vol. 7, no. 1, pp. 77 91, 1952. D. Palomar (HKUST) CVaR Portfolio 15 / 58

Mean-Variance Tradeoff The mean return w T µ and the variance (risk) w T Σw constitute two important performance measures. Usually, the higher the mean return the higher the variance and vice-versa. Thus, we are faced with two objectives to be optimized: it is a multi-objective optimization problem. They define a fundamental mean-variance tradeoff curve (Pareto curve). The choice of a specific point in this tradeoff curve depends on how agressive or risk-averse the investor is. D. Palomar (HKUST) CVaR Portfolio 16 / 58

Markowitz mean-variance portfolio (1952) The idea of the Markowitz framework 5 is to find a trade-off between the expected return w T µ and the risk of the portfolio measured by the variance w T Σw: maximize w w T µ λw T Σw subject to 1 T w = 1 where w T 1 = 1 is the capital budget constraint and λ is a parameter that controls how risk-averse the investor is. This is a convex QP with only one linear constraint which admits a closed-form solution: w = 1 2λ Σ 1 (µ + ν 1), where ν is the optimal dual variable ν = 2λ 1T Σ 1 µ. 1 T Σ 1 1 5 H. Markowitz, Portfolio selection, J. Financ., vol. 7, no. 1, pp. 77 91, 1952. D. Palomar (HKUST) CVaR Portfolio 17 / 58

Global Minimum Variance Portfolio (GMVP) The global minimum variance portfolio (GMVP) ignores the expected return and focuses on the risk only: minimize w w T Σw subject to 1 T w = 1. It is a simple convex QP with solution w GMVP = 1 1 T Σ 1 1 Σ 1 1. It is widely used in academic papers for simplicity of evaluation and comparison of different estimators of the covariance matrix Σ (while ignoring the estimation of µ). D. Palomar (HKUST) CVaR Portfolio 18 / 58

Drawbacks of Markowitz s formulation The Markowitz portfolio has never been embraced by practitioners, among other reasons because 1 variance is not a good measure of risk in practice since it penalizes both the unwanted high losses and the desired low losses: the solution is to use alternative measures for risk, e.g., VaR and CVaR, 2 it is highly sensitive to parameter estimation errors (i.e., to the covariance matrix Σ and especially to the mean vector µ): solution is robust optimization, 3 it only considers the risk of the portfolio as a whole and ignores the risk diversification: solution is the risk-parity portfolio. We will here consider more meaningful measures for risk than the variance, like the downside risk (DR), Value-at-Risk (VaR), Conditional VaR (CVaR) or Expected Shortfall (ES), and drawdown (DD). D. Palomar (HKUST) CVaR Portfolio 20 / 58

Variance as risk measure In finance, the mean return is very relevant as it quantifies the average benefit of the investment. However, in practice, the average performance is not good enough and one needs to control the probability of going bankrupt. Risk measures control how risky an investment strategy is. The most basic measure of risk is the variance as considered by Markowitz in 1952: 6 a higher variance means that there are large peaks in the risk distribution which may cause a big loss. However, Markowitz himself already recognized and stressed the limitations of the mean-variance analysis. 7 1959. 6 H. Markowitz, Portfolio selection, J. Financ., vol. 7, no. 1, pp. 77 91, 1952. 7 H. Markowitz, Portfolio Selection: Efficient Diversification of Investments. Wiley, D. Palomar (HKUST) CVaR Portfolio 22 / 58

Alternatives to variance as risk measure Variance is not a good measure of risk in practice since it penalizes both the unwanted high losses and the desired low losses (or gains). 8 Indeed, the mean-variance portfolio framework penalizes up-side and down-side risk equally, whereas most investors don t mind up-side risk. To overcome the limitations of the variance as risk measure, a number of alternative risk measures have been proposed, for example: Downside Risk (DR) Value-at-Risk (VaR) Conditional Value-at-Risk (CVaR) Drawdown (DD): maximum DD average DD Conditional Drawdown at Risk (CDaR) 8 A. McNeil, R. Frey, and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, 2005. D. Palomar (HKUST) CVaR Portfolio 23 / 58

Downside Risk (DR) Let R be a random variable representing the return of an asset or portfolio (e.g., R = w T r where r denotes the vector of random returns of the assets). We are familiar with the mean return µ = E[R] and with the variance σ 2 = E[(R µ) 2 ]. The idea of downside risk is that the left-handside of the return distribution involves risk while the right-handside contains the better investment opportunities. Interest in downside risk arose in the early 1950s. One example is the semi-variance, already considered by Markowitz in 1959. 9 The semi-variance measures the variability of the returns below the mean. 1959. 9 H. Markowitz, Portfolio Selection: Efficient Diversification of Investments. Wiley, D. Palomar (HKUST) CVaR Portfolio 24 / 58

LPM and semivariance The semivariance is a special case of the more general lower partial moments (LPM): LPM = E [ ((τ R) + ) α ], where ( ) + = max(0, ). The parameter τ is termed the disaster level. The parameter α reflects the investor s feeling about the relative consequences of falling short of τ by various amounts: the value α = 1 (which suits a neutral investor) separates risk-seeking (0 < α < 1) from risk-averse (α > 1) behavior with regard to returns below the target τ. By changing the parameters α and τ most downside measures used in practice can be formed. In particular, setting α = 2 and τ = E[R] yields the semi-variance (or lower partial variance): SV = E [ ((E[R] R) + ) 2 ]. D. Palomar (HKUST) CVaR Portfolio 25 / 58

Value-at-Risk (VaR) To overcome the drawback of variance, another popular single side risk measurement is the Value-at-Risk (VaR) initially proposed by J.P. Morgan. VaR denotes the maximum loss with a specified confidence level (e.g., confidence level = 95%, period = 1 day). Let ξ be a random variable representing the loss from a portfolio over some period of time (e.g., ξ = w T r where r denotes the vector of random returns of the assets). The VaR is defined as VaR α = inf {ξ 0 : Pr (ξ ξ 0 ) α} with α the confidence level, say, α = 0.95. However, this measure does not take into account losses exceeding VaR, is nonconvex, and is not subadditive. D. Palomar (HKUST) CVaR Portfolio 26 / 58

Conditional Value-at-Risk (CVaR) The Conditional Value-at-Risk (CVaR) is also called Expected Shortfall (ES). The CVaR takes into account the shape of the losses exceeding the VaR through the average: CVaR α = E [ξ ξ VaR α ]. D. Palomar (HKUST) CVaR Portfolio 27 / 58

Drawdown The drawdown (DD) at time t is defined as the decline from a historical peak of the cumulative profit X(t). The unnormalized version is D(t) unnorm = max X(τ) X(t). 1 τ t But in practice, the normalized version is used: D(t) = HWM t X(t) HWM t where HWM t is the high water mark of X(t) defined as HWM t = max 1 τ t X(τ). D. Palomar (HKUST) CVaR Portfolio 28 / 58

Drawdown S&P 500 Cumulative Return 2010 01 05 / 2015 12 30 1.6 1.4 1.2 1.0 Daily Return 0.04 0.02 0.00 0.02 0.04 0.06 0.05 0.10 0.15 0.20 Drawdown Jan 05 2010 Jul 01 2010 Jan 03 2011 Jul 01 2011 Jan 03 2012 Jul 02 2012 Jan 02 2013 Jul 01 2013 Jan 02 2014 Jul 01 2014 Jan 02 2015 Jul 01 2015 Dec 30 2015 D. Palomar (HKUST) CVaR Portfolio 29 / 58

Drawdown Then one can define the maximum DD (Max-DD) over a period t = 1,..., T as M(T) = max 1 t T D(t) Also the average DD (Ave-DD) over a period t = 1,..., T as A(T) = 1 T 1 t T D(t) Similarly to the CVaR, we can define the Conditional Drawdown at Risk (CDaR) as the mean of the worst 100(1 α)% drawdowns. D. Palomar (HKUST) CVaR Portfolio 30 / 58

Mean-downside risk portfolio Recall Markowitz mean-variance portfolio formulation: maximize w w T µ λw T Σw subject to 1 T w = 1, w 0. Instead of using the variance we can use a downside risk measure, obtaining the mean-downside risk formulation (introduced in 1977). For example, the LPM can be approximated as [ ((τ E R) + ) α ] 1 T T ( (τ Rt ) +) α where R t = w T r t. The mean-lpm portfolio formulation is the convex (depending on α) problem ( maximize w T µ λ 1 ( ) ) Tt=1 + α w T τ w T r t subject to 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 32 / 58 t=1

Mean-semivariance portfolio In particular, we can approximate the semivariance as [ ((E[R] E R) + ) 2 ] 1 T 1 T T ((E[R] R t ) +) 2 t=1 ( T 1 t=1 T ) + 2 T R t R t The mean-semivariance portfolio formulation is the convex QP problem t=1 ( maximize w T µ λ 1 ( ) ) Tt=1 + 2 w T w T µ w T r t subject to 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 33 / 58

Mean-LPM portfolio portfolio with different α s For less risk-averse investors, we can consider the mean-lpm portfolio formulation with α = 1, which is an LP: maximize w T µ λ 1 ) Tt=1 + w T (w T µ w T r t subject to 1 T w = 1, w 0. For more risk-averse investors, we can consider the mean-lpm convex portfolio formulation with α = 3: ( maximize w T µ λ 1 ( ) ) Tt=1 + 3 w T w T µ w T r t subject to 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 34 / 58

Mean-CVaR portfolio A portfolio formulation dealing directly with VaR and CVaR quantities is not tractable. Let f (w, r) be an arbitrary cost function, where w is the optimization variable (portfolio) and r denotes the random asset returns. Example: f (w, r) = w T r. Consider, for example, the maximization of the mean return subject to a CVaR risk constraint on the loss: maximize w subject to w T µ CVaR α (f (w, r)) c 1 T w = 1, w 0 where CVaR α (f (w, r)) = E [f (w, r) f (w, r) VaR α (f (w, r))]. D. Palomar (HKUST) CVaR Portfolio 36 / 58

CVaR portfolio Rockafellar and Uryasev 10 first proposed to minimize the CVaR of the portfolio loss as follows: ) minimize CVaR α (w T r w subject to 1 T w = 1, w 0 where ) ( )] CVaR α (w T r = E [w T r w T r VaR α w T r. 10 R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, J. Risk, vol. 2, pp. 21 42, 2000. D. Palomar (HKUST) CVaR Portfolio 37 / 58

CVaR in Convex Form Define the auxiliary convex function where [x] + = max (x, 0). F α (w, ζ) = ζ + 1 1 α E[ wt r ζ] +, Rockafellar and Uryasev show that 1 VaR α ( w T r) is a minimizer of F α (w, ζ) w.r.t. ζ: VaR α ( w T r) arg min F α (w, ζ). ζ 2 CVaR α ( w T r) equals minimum F α (w, ζ) w.r.t. ζ: CVaR α ( w T r) = min F α (w, ζ). ζ D. Palomar (HKUST) CVaR Portfolio 38 / 58

Proof CVaR in Convex Form 1 The minimizer of F α (w, ζ) w.r.t. ζ satisfies: 0 ζ F α (w, ζ ). For example, we choose the following subgradient: 0 = s ζ F α (w, ζ ) = 1 1 1 1 α { w T r>ζ }p(r)dr = 1 1 ( 1 α P w T r > ζ ), where 1 is the indicator function. Solving the above equation, we have ( P w T r > ζ ) = 1 α = ζ = VaR α ( w T r). D. Palomar (HKUST) CVaR Portfolio 39 / 58

Proof CVaR in Convex Form 2 First, we have min ζ F α (w, ζ) = F α (w, ζ ) = ζ + 1 1 α E[ wt r ζ ] +. Recall that [ CVaR α ( w T r) = E w T r ] w T r > VaR α ( w T r) = 1 ( ) w T r p(r)dr 1 α w T r>var α( w T r) = 1 [ ] + w T r VaR α ( w T r) p(r)dr 1 α + VaR α ( w T r). D. Palomar (HKUST) CVaR Portfolio 40 / 58

CVaR in Convex Form Corollary: ( ) min CVaR α w T r = min F α (w, ζ) w w,ζ In words, minimizing F α (w, ζ) simultaneously calculates the optimal CVaR and VaR. Corollary: Because w T r is convex in w for each r, then F α (w, ζ) is convex! Proof: F α (w, ζ) = ζ + 1 [ ] + w T r ζ p (r) dr. 1 α D. Palomar (HKUST) CVaR Portfolio 41 / 58

Sample Average Approximation of CVaR Sample average approximation of F α (w, ζ): F α (w, ζ) = ζ + 1 1 α E[ wt r ζ] + ζ + 1 1 1 α T T [ w T r t ζ] +. t=1 D. Palomar (HKUST) CVaR Portfolio 42 / 58

CVaR portfolio as an LP We first include the dummy variables z t : z t [ w T r t ζ] + = zt w T r t ζ, z t 0 CVaR portfolio problem can be approximated by an LP: minimize w,{z t},ζ ζ + 1 1 Tt=1 1 α T z t subject to 0 z t w T r t ζ, t = 1,..., T 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 43 / 58

Mean-CVaR portfolio as an LP We can also consider the maximization of the mean return subject to a CVaR constraint: maximize w,{z t},ζ w T µ subject to ζ + 1 Or a mean-cvar objective: maximize w,{z t},ζ 1 1 α Tt=1 T z t c 0 z t w T r t ζ, t = 1,..., T 1 T w = 1, w 0. ( w T µ λ ζ + 1 1 1 α T Tt=1 z t ) subject to 0 z t w T r t ζ, t = 1,..., T 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 44 / 58

Drawdown (DD) Let r(t) be the return vector of the N stocks at time t. Define the cumulative (uncompounded) return vector as r cum (t) = t r(τ) τ=1 (Note: the compounded return is t τ=1 (1 + r(τ)) 1.) The portfolio return is r p (t) = w T r(t) and cumulative return r cum p (t) = w T r cum (t) The drawdown (DD) at time t can be written as D(t) = max 1 τ t rcum p (τ) r cum (t) p D. Palomar (HKUST) CVaR Portfolio 46 / 58

Max-DD, Ave-DD, and CDaR The maximum DD (Max-DD) over a period t = 1,..., T is M(T) = max 1 t T D(t) The average DD (Ave-DD) over a period t = 1,..., T is A(T) = 1 T 1 t T D(t) Similarly to the CVaR, we can define the Conditional Drawdown at Risk (CDaR) as the mean of the worst 100(1 α)% drawdowns: α (T) = 1 (1 α)t t Ω α D(t), where Ω α = {1 t T D(t) ξ α } with ξ α being the threshold such that exactly 100(1 α)% of drawdowns exceeds that limit. D. Palomar (HKUST) CVaR Portfolio 47 / 58

CDaR in Convex Form The CDaR can be conveniently expressed as 11 α (w) = min{ζ + 1 ζ 1 α 1 T T [D t (w) ζ] + } When α tends to 1, the CDaR tends to the maximum drawdown, i.e., t=1 1 (T) = M(T) When α tends to 0, the CDaR tends to the average drawdown, i.e., 0 (T) = A(T) 11 A. Chekhlov, S. Uryasev, and M. Zabarankin, Portfolio optimization with drawdown constraints, Research Report 2000-5. Available at SSRN: https://ssrn.com/abstract=223323orhttp://dx.doi.org/10.2139/ssrn.223323, 2000. D. Palomar (HKUST) CVaR Portfolio 48 / 58

Mean-Max-DD portfolio as an LP We can consider the maximization of the mean return subject to a Max-DD constraint: maximize w subject to w T µ max 1 t T {max 1 τ t w T r cum τ 1 T w = 1, w 0. w T r cum t } c Removing one maximum operator is trivial: maximize w subject to w T µ max 1 τ t w T r cum τ w T r cum t c, 1 t T 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 49 / 58

Mean-Max-DD portfolio as an LP To remove the other max operator, we need to introduce some additional variables: maximize w,{u t} subject to w T µ u t w T r cum t c, 1 t T u t w T r cum τ 1 t T, 1 τ t 1 T w = 1, w 0. We can reduce the large number of constraints by rewriting it as maximize w,{u t} subject to w T µ u t w T r cum t c, 1 t T u t w T r cum t u t u t 1 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 50 / 58

Mean-Max-DD portfolio as an LP We can finally write the maximization of the mean return subject to the Max-DD constraint as maximize w,{u t} subject to w T µ w T r cum t u t w T r cum t + c, 1 t T u t 1 u t 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 51 / 58

Mean-Ave-DD portfolio as an LP Similarly, we can consider the maximization of the mean return subject to an Ave-DD constraint: or, equivalently, maximize w,{u t} 1 subject to w T µ T Tt=1 (u t w T r cum t ) c u t w T r cum t u t 1 u t 1 T w = 1, w 0 maximize w T µ w,{u t} 1 subject to Tt=1 T u t T t=1 w T r cum t w T r cum t u t u t 1 u t 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 52 / 58 + c

Mean-CDaR portfolio as an LP Finally, we can consider the maximization of the mean return subject to a CDaR constraint: maximize w,ζ w T µ subject to ζ + 1 1 1 α T 1 T w = 1, w 0. Tt=1 [max 1 τ t w T r cum τ w T r cum t ζ] + c Similarly to the CVaR case, we can get rid of the [ ] + operator by introducing some additional variables: maximize w,{z t},ζ w T µ subject to ζ + 1 1 1 α Tt=1 T z t c 0 z t max 1 τ t w T r cum τ 1 T w = 1, w 0. w T r cum t ζ, t = 1,..., T D. Palomar (HKUST) CVaR Portfolio 53 / 58

Mean-CDaR portfolio as an LP Similarly to the Max-DD and Ave-DD cases, we can get rid of the max operators by introducing additional variables: maximize w,{z t},ζ,{u t} w T µ subject to ζ + 1 1 Tt=1 1 α T z t c 0 z t u t w T r cum t ζ, t = 1,..., T w T r cum t u t u t 1 u t 1 T w = 1, w 0. D. Palomar (HKUST) CVaR Portfolio 54 / 58

Word of caution on DD The maximum drawdown is extremely sensitive to minute changes in the portfolio weights and to the specific time period examined. If returns are close to normally distributed, the distribution of drawdowns is just a function of the variance, so there s no need to include drawdowns explicitly in your portfolio construction objective. Minimizing variance is the same as minimizing expected drawdowns. On the other hand, if returns are very non-normal and you want to find a portfolio that minimizes the expected drawdowns, you still wouldn t choose weights that minimize historical drawdown. Why? Because minimizing historical drawdown is effectively the same as taking all your returns that weren t part of a drawdown, and hiding them from your optimizer, which will lead to portfolio weights that are a lot less accurately estimated than if you let your optimizer see all the data you have. Instead, you might just include terms in your optimization objective that penalize negative skew and penalize positive kurtosis. D. Palomar (HKUST) CVaR Portfolio 55 / 58

Conclusions We have reviewed the Markowitz portfolio formulation and understood that it has many practical flaws that make it impractical. Indeed, it is not used by practitioners. We have learned about alternative measures of risk as opposed to variance: Downside risk (one particular example is the semi variance) VaR CVaR Drawdown We have formulated several portfolio designs based on downside risk, CVaR, and DD, all as LPs! D. Palomar (HKUST) CVaR Portfolio 57 / 58

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